Math
Rørdam K-theory
2.2.10 Liftings
Chapter 2 Projections and Unitary elements
Chapter 3 The K_0 Group of a Unital C -algebra.
Chapter 4 The functor K0
Chapter 5 The ordered abelian group K0
Chapter 6 Inductive limits
Characterization for stable equivalence
Corollary 2.1.4
Definition 2.1.2.
Definition 2.2.1
Definition 2.3.1 (The semigroup of projections of A)
Definition 2.3.3. (The semigroup of projections)
Definition 3.1.4 (The K_0 group of a Unital C-algebra)
Definition 3.1.6 (Stable equivalence)
Definition 3.2.5 (Homotopy equivalence)
Definition 4.1.1 (K0, non-unital case)
Definition 4.2.1 (The Scalar mapping)
Definition 5.1.1.
Definition 5.1.3
Definition 5.1.4 (Positive cone of K0)
Definition 5.16
Definition 6.4.a (Stabilization)
Drawing 2024-02-05 22.07.46
DRAWING
Example 3.1.3
Example 3.3.2
Example 3.3.3
Example 3.3.5
Example 4.1.5
Example 4.3.5
Lemma 2.1.3
Lemma 2.1.5 (Whitehead)
Lemma 2.1.7.
Lemma 2.2.3
Lemma 3.2.7
Lemma 3.2.8
Lemma 4.3.2
Lemma 5.1.2
Lemma 6.3.1
Murray-von Neumann equivalence
Paragraph 3.3.1 Traces and K0
Properties of the Grothendieck group
Proposition 2.1.6.
Proposition 2.1.8
Proposition 2.1.11
Proposition 2.2.5
Proposition 2.2.6
Proposition 2.2.7
Proposition 2.2.8
Proposition 2.3.2.
Proposition 3.1.7 (The standard picture for K0 — the unital case)
Proposition 3.1.8 (Universal property of K0)
Proposition 3.2.4 (Functoriality of K0 for unital C-star Algebras)
Proposition 3.2.6 (Homotopy invariance of K0)
Proposition 4.1.3 (Functoriality of K0)
Proposition 4.1.4 (Homotopy invariance of K0)
Proposition 4.3.2 (Half exactness of K0)
Proposition 4.3.3 (Split exactness of K0)
Proposition 4.3.4 (Direct sums)
Proposition 4.3.8 (Stability)
Proposition 5.1.5
Proposition 5.1.7
Proposition 5.1.9.
Proposition 6.2.4
Proposition 6.4.1 (Stability of K0)
retract
Rørdam K-theory
The functor K00 3.2.3
The functor K0 for unital C-star Algebras 3.2.2
The Grothendieck construction 3.1.1
The group K_00 3.1.5
Theorem 2.1.9. (The polar decomposition)
Theorem 2.1.10 (Carl Neumann Series)
Theorem 4.1.2. (The functor K0)
Theorem 5.2.1
Theorem 5.2.2
Theorem 6.3.2 (Continuity of K0)
Diffuse measure
dirac measures
finite C-star algebra
finite projection
homotopy
infinite C-star algebra
infinite projection
probability measure
quasi-trace
stably finite

Chapter 5 The ordered abelian group K0

An extra structure is added to the abelian group of a -algebra by specifying a certain subset of it, called . The set consists of all elements in of the form for some projection in when is unital, and stably finite, then has the pleasant structure of an ordered abelian group. We shall for this purpose also discuss finiteness properties of -algebras and of projections.

5.1 The ordered K0-group of stably finite C^* algebras

Definition 5.1.1.
A projection in a -algebra is said to be an infinite projection if it is Murray-von Neumann equivalent to a proper su projection of itself. In symbols, if such that . If is not an infinite projection then we say that is a finite projection. A unital -algebra is said to be a finite C-star algebra if its unit is a finite projection . Otherwise we say that is an infinite C-star algebra . If is a finite C-star algebra for all , then we say that is stably finite.

NOTE: A projection in a -algebra is a finite projection if and only if is a finite C-star algebra.

Lemma 5.1.2
The following conditions are equivalent for any unital -algebra

  1. is a finite C-star algebra.
  2. All isometries in are unitary.
  3. All projections in are finite projections.
  4. Every left-invertible element in is invertible.
  5. Every right-invertible element in is invertible.

Proof:
()
If is an isometry, then by direct calculation we know that is a projection. Hence and is a unitary.
()
If for some projection . Then by the definition of murray von neumann equivalence, there exists an isometry which connects and . Thus and .But in this algebra, every isometry is a unitary, so , this shows that is equivalent to a proper subprojection of itself.
()
Suppose that and are two projections in such that and . Let be a partial isometry in which implements the equivalence between and . and . Put and from our assumption, so we get that . We conclude that and , but since all isometries are unitary, necessarily, proving that every projection is equivalent to some proper sub projection of itself.
() If holds, then the unit is a finite projection.
() Parse through what it means.
() Trivial.
()
Suppose that holds, and let be a left-invertible element in . find such that . Recall that if are self-adjoint elements in a unital -algebra , then and if then for each . Use these facts to obtain the inequality This shows that and the spectrum of is therefore contained in the interval , which in particular implies tat is invertible. Put , and observe that therefore is invertible by (2), and hence is invertible.
Warnings:

  • A finite -algebra, unital or not, will always satisfy (3), that all projections are finite.
    The converse is NOT TRUE
  • A non-unital -algebra can be infinite and satisfy (3), have only finite projections.
  • Lastly, a finite C-star algebra does not have to be stably finite.

Definition 5.1.3
A pair is called an ordered abelian group if is an abelian Group, is a subset of and Define a relation on by if .

Definition 5.1.4 (Positive cone of K0)
for a -algebra , the positive cone of is

Proposition 5.1.5
Let be a -algebra.

  2. if is unital, then
  3. If is stably finite, then
  4. If is unital and stably finite then is an ordered abelian group.

NOTE:
Let be a properly infinite, unital -algebra, with nontrivial -group; the cuntz algebras for are examples of such algebras. Then by exercise 4.6(4), and hence therefore, are not ordered abelian groups.

Definition 5.16
An element in an ordered abelian group is called an order unit if for ever , there exists such that . In other words, you can wrangle any element with this order unit in the same way you can wrangle any number in the integers with 1. It makes sense I promise.

Proposition 5.1.7
If is a unital -algebra, then is an order unit for .
Proof:
Let be given. By the standard picture of of a unital -algebra we can find projections for some such that . Let denote the unit of , and write instead of . Then and belong to . Hence

Proposition 5.1.9.
The positive cone of is given by

5.2 States on K0(A) and traces on A

Suppose that is an ordered abelian group, with a distinguished order unit. A state on is a group homomorphism satisfying and . i.e. a state is an order preserving, positive group homomorphism from to . The set of all states on is denoted or just .

The following structure on can partly be recovered from its state space as the following Hahn-Banach type theorem by K. Goodearl, and D. Handelman shows.

Theorem 5.2.1
Let be an ordered abelian group with distinguished order unit , and let . Then for all states on if and only if is an order unit for for some positive integer

Theorem 5.2.2
Every state on , where is a unital -algebra, is of the form for some quasi-trace on .
If is an exact C-star algebra then is a trace.

Table Of Contents
Chapter 5 The ordered abelian group K0
5.1 The ordered K0-group of stably finite C^* algebras
5.2 States on K0(A) and traces on A